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Sagot :
To create a table of values for the function [tex]\( f(x) = \frac{1}{4} |x| \)[/tex], we need to choose specific values of [tex]\( x \)[/tex] and then determine the corresponding [tex]\( f(x) \)[/tex] values by plugging them into the equation. Here, we will consider a symmetric range of [tex]\( x \)[/tex] values for simplicity. Let's choose [tex]\( x = -10, -5, 0, 5, 10 \)[/tex].
Let's calculate the values step-by-step:
1. For [tex]\( x = -10 \)[/tex]:
[tex]\[ f(-10) = \frac{1}{4} \cdot |-10| = \frac{1}{4} \cdot 10 = 2.5 \][/tex]
2. For [tex]\( x = -5 \)[/tex]:
[tex]\[ f(-5) = \frac{1}{4} \cdot |-5| = \frac{1}{4} \cdot 5 = 1.25 \][/tex]
3. For [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = \frac{1}{4} \cdot |0| = \frac{1}{4} \cdot 0 = 0.0 \][/tex]
4. For [tex]\( x = 5 \)[/tex]:
[tex]\[ f(5) = \frac{1}{4} \cdot |5| = \frac{1}{4} \cdot 5 = 1.25 \][/tex]
5. For [tex]\( x = 10 \)[/tex]:
[tex]\[ f(10) = \frac{1}{4} \cdot |10| = \frac{1}{4} \cdot 10 = 2.5 \][/tex]
Now we can present these values in a table:
\begin{tabular}{|c|c|c|c|c|c|}
\hline
[tex]\( x \)[/tex] & -10 & -5 & 0 & 5 & 10 \\
\hline
[tex]\( f(x) \)[/tex] & 2.5 & 1.25 & 0.0 & 1.25 & 2.5 \\
\hline
\end{tabular}
Graphing the Function:
1. Plot the Points: Use the table of values above to plot the points [tex]\((-10, 2.5)\)[/tex], [tex]\((-5, 1.25)\)[/tex], [tex]\((0, 0.0)\)[/tex], [tex]\((5, 1.25)\)[/tex], and [tex]\((10, 2.5)\)[/tex] on a coordinate plane.
2. Draw the Curve: Connect these points smoothly, remembering that [tex]\( f(x) = \frac{1}{4} |x| \)[/tex] forms a V-shaped graph because it is an absolute value function. The vertex of the V is at the origin [tex]\((0, 0.0)\)[/tex]. The lines are symmetric around the y-axis.
When you graph these points and connect them properly, you will see the characteristic shape of the absolute value function scaled vertically by a factor of [tex]\(\frac{1}{4}\)[/tex].
Let's calculate the values step-by-step:
1. For [tex]\( x = -10 \)[/tex]:
[tex]\[ f(-10) = \frac{1}{4} \cdot |-10| = \frac{1}{4} \cdot 10 = 2.5 \][/tex]
2. For [tex]\( x = -5 \)[/tex]:
[tex]\[ f(-5) = \frac{1}{4} \cdot |-5| = \frac{1}{4} \cdot 5 = 1.25 \][/tex]
3. For [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = \frac{1}{4} \cdot |0| = \frac{1}{4} \cdot 0 = 0.0 \][/tex]
4. For [tex]\( x = 5 \)[/tex]:
[tex]\[ f(5) = \frac{1}{4} \cdot |5| = \frac{1}{4} \cdot 5 = 1.25 \][/tex]
5. For [tex]\( x = 10 \)[/tex]:
[tex]\[ f(10) = \frac{1}{4} \cdot |10| = \frac{1}{4} \cdot 10 = 2.5 \][/tex]
Now we can present these values in a table:
\begin{tabular}{|c|c|c|c|c|c|}
\hline
[tex]\( x \)[/tex] & -10 & -5 & 0 & 5 & 10 \\
\hline
[tex]\( f(x) \)[/tex] & 2.5 & 1.25 & 0.0 & 1.25 & 2.5 \\
\hline
\end{tabular}
Graphing the Function:
1. Plot the Points: Use the table of values above to plot the points [tex]\((-10, 2.5)\)[/tex], [tex]\((-5, 1.25)\)[/tex], [tex]\((0, 0.0)\)[/tex], [tex]\((5, 1.25)\)[/tex], and [tex]\((10, 2.5)\)[/tex] on a coordinate plane.
2. Draw the Curve: Connect these points smoothly, remembering that [tex]\( f(x) = \frac{1}{4} |x| \)[/tex] forms a V-shaped graph because it is an absolute value function. The vertex of the V is at the origin [tex]\((0, 0.0)\)[/tex]. The lines are symmetric around the y-axis.
When you graph these points and connect them properly, you will see the characteristic shape of the absolute value function scaled vertically by a factor of [tex]\(\frac{1}{4}\)[/tex].
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